Necessary and sufficient conditions of a class of bipartite graphs with local antimagic chromatic number 2 - an algebraic approach

For a connected graph $G = (V, E)$, a bijective edge labeling $f:E \to\{1,\ldots ,|E|\}$ is a local antimagic labeling of $G$ if it induces a vertex labeling $f^+$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label $f^+(x)= \sum f(xu)$, with $u$ ranging over all the vertices adjacent to $x$. The minimum number of distinct induced vertex labels over all local antimagic labelings of $G$ is the local antimagic chromatic number of $G$, denoted $χ_{la}(G)$. In this paper, we make use of algebraic analysis to obtain necessary and sufficient conditions for every bipartite graph with all vertices of degree 2 except exactly three vertices of degree at least 3 to have local antimagic chromatic number 2. Moreover, we showed that the consecutive edge labels of every induced path of each case is unique.